Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-8k^2 - 88k - 144}{-3k^3 - 21k^2 + 54k}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-8(k^2 + 11k + 18)} {-3k(k^2 + 7k - 18)} $ $ y = \dfrac{8}{3k} \cdot \dfrac{k^2 + 11k + 18}{k^2 + 7k - 18} $ Next factor the numerator and denominator. $ y = \dfrac{8}{3k} \cdot \dfrac{(k + 9)(k + 2)}{(k + 9)(k - 2)}$ Assuming $k \neq -9$ , we can cancel the $k + 9$ $ y = \dfrac{8}{3k} \cdot \dfrac{k + 2}{k - 2}$ Therefore: $ y = \dfrac{ 8(k + 2)}{ 3k(k - 2)}$, $k \neq -9$